On the convergence of sequences of positive linear operators towards composition operators

The main aim of the paper is to investigate some sufficient conditions which guarantee the convergence of sequences of positive linear operators towards composition operators within the framework of function spaces defined on a metric space. Among other things, the adopted approach allows to obtain a unifying reassessment of two milestones of the approximation theory by positive linear operators, namely, Korovkin’s theorem and Feller’s theorem together with some new extensions of them to the more general case where the limit operator is a composition operator. Some applications are shown and, among them, the convergence of Bernstein–Schnabl operator is enlightened in the framework of Banach spaces.